Finite quotients of the multiplicative group of a finite dimensional division algebra are solvable |
| |
Authors: | Andrei S. Rapinchuk Yoav Segev Gary M. Seitz |
| |
Affiliation: | Department of Mathematics, University of Virginia, Charlottesville, Virginia 22904 ; Department of Mathematics, Ben-Gurion University, Beer-Sheva 84105, Israel ; Department of Mathematics, University of Oregon, Eugene, Oregon 97403-1226 |
| |
Abstract: | We prove that finite quotients of the multiplicative group of a finite dimensional division algebra are solvable. Let be a finite dimensional division algebra having center , and let be a normal subgroup of finite index. Suppose is not solvable. Then we may assume that is a minimal nonsolvable group (MNS group for short), i.e. a nonsolvable group all of whose proper quotients are solvable. Our proof now has two main ingredients. One ingredient is to show that the commuting graph of a finite MNS group satisfies a certain property which we denote Property . This property includes the requirement that the diameter of the commuting graph should be , but is, in fact, stronger. Another ingredient is to show that if the commuting graph of has Property , then is open with respect to a nontrivial height one valuation of (assuming without loss of generality, as we may, that is finitely generated). After establishing the openness of (when is an MNS group) we apply the Nonexistence Theorem whose proof uses induction on the transcendence degree of over its prime subfield to eliminate as a possible quotient of , thereby obtaining a contradiction and proving our main result. |
| |
Keywords: | Division algebra multiplicative group finite homomorphic images valuations |
|
| 点击此处可从《Journal of the American Mathematical Society》浏览原始摘要信息 |
|
点击此处可从《Journal of the American Mathematical Society》下载全文 |
|